We present a new way to deal with doubly in nite lower Hessenberg matrices based on the representation of the matrices as the sum of their diagonal submatrices. We show that such representation is a simple and useful tool for computation purposes and also to obtain general properties of the matrices related with inversion, similarity, commutativity, and Pincherle derivatives. The diagonal representation allows us to consider the ring of doubly in nite lower Hessenberg matrices over a ring R as a ring of Laurent series in one indeterminate, with coe cients in the ring of R-valued sequences that don't commute with the indeterminate.
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